Imperfect Information Transmission from Banks to Investors:
Macroeconomic Implications
Nicolás Figueroa, Oksana Leukhina, Carlos Ramírez
August 12, 2016
We study the interaction of information transmission in loan-backed asset markets and screening
e¤ort at origination. Originating banks can screen their borrowers, but can inform investors of their
asset type only through an imperfect rating technology. The premium paid on highly rated assets
emerges as the main determinant of screening e¤ort. Because the rating technology is imperfect,
the premium is insu¢ cient to induce an e¢ cient level of screening. Rising collateral values and
increasing asset complexity help explain several key pre-crisis trends: (1) laxer screening standards,
(2) intensi…ed ratings shopping, (3) ratings in‡ation, and (4) the decline in the premium paid on
highly rated assets. Contrary to conventional wisdom, we …nd that mandatory rating and mandatory
ratings disclosure policies exacerbate the credit misallocation problem.
JEL Codes: G01, G24, G28
Keywords: credit misallocation, information asymmetry, screening, loan sales, ratings
Acknowledgements: We would like to thank Philip Bond, Theo Eicher, Mark Flannery, Kristopher Gerardi, Burton
Holli…eld, Fahad Khalil, Markus Opp, Bob Richards, Uday Rajan, William Roberds, Bryan Routledge, Francesco Sangiorgi, Chester Spatt, Venky Venkateswaran and the seminar participants at the University of Washington, Ponti…cia
Universidad Católica de Chile, the Federal Reserve Bank of Atlanta, the 2013 Midwest Macro Conference, the 2014
Midwest Economics Association Conference, the 2014 North American Meetings of the Econometric Society, the 2015
Carnegie Mellon University Conference organized by Francesco Sangiorgi and Chester Spatt (“The Economics of Credit
Rating Agencies, Credit Ratings and Information Intermediaries”).
Nicolas Figueroa : Instituto de Economía, Ponti…cia Universidad Católica de Chile; Oksana Leukhina: University of Washington; Carlos Ramírez : Carnegie Mellon University
1.
Introduction
The …ve year economic expansion leading up to the 2007-2008 …nancial crisis witnessed an unprecedented
growth of markets for securitized products. The ten-fold growth in the annual issue of loan-backed securities
accounts for roughly half of this expansion, and it will be the focus of this paper for concreteness [Bord and
Santos (forthcoming)].1 Several empirical papers, e.g. Purnanandam (2011), Keys, Mukherjee, Seru, and Vig
(2010), Keys, Seru, and Vig (2012), Bord and Santos (forthcoming), suggested that the rise of securitization
directly contributed to laxer screening standards observed prior to the crisis, lending support to economists’
public opinion regarding the adverse consequences of securitization on the originators’incentives to screen their
borrowers [Stiglitz (2007), Blinder (2007)].2
Nonetheless, our theoretical understanding of the real implications of markets for loan-backed assets remains limited [Gorton and Metrick (2011)]. Our contribution is to investigate the macroeconomic implications
of information asymmetry present in these markets between banks, whose screening choices impact the economy,
and investors, who provide the funds and bear the risk.3 Much of the theoretical e¤ort in this …eld, exempli…ed
by Gorton and Pennacchi (1995) and Parlour and Plantin (2008), has aimed to explain the existence of these
markets, which seem to interfere with the classic role of intermediaries discussed in Boyd and Prescott (1986)
and Holmstrom and Tirole (1997). This literature largely agrees on liquidity needs as underlying existence of
these markets. We are not interested in explaining the presence of these markets. Therefore, building on these
insights, we assume that banks have no funds and must raise them through loan sales, which e¤ectively ensures
the presence of markets for loans.
We develop a general equilibrium model with heterogeneous borrowers, heterogeneous banks, and representative investors. In order to raise funds, banks must sell their baskets of loans to investors. Consistent with
the classic role of intermediaries, banks are able to make sure, at a cost, that they extend a high quality loan
basket. The information available to banks about their asset is not observable by investors, but an imperfect
rating technology, which reveals the true asset type with a …xed probability, is available to banks at a cost.
Pro…t maximizing banks choose whether or not to screen, rate, and disclose the ratings. Employing the rating
technology should be interpreted loosely as engaging in a costly process, which results, with some positive
probability, in the enhancement of the perceived value of the asset.
We then employ this framework to investigate how endogenous information production in markets for
loans interacts with the banks’screening e¤ort at the stage of loan origination. The following important insight
emerges from the model: It is the premium paid on assets with good ratings that disciplines the screening e¤ort.
Therefore, understanding what determines this premium is crucial for understanding the credit allocation in
the economy.
Our analysis reveals that the premium paid on highly rated assets is directly proportional to an endogenous
object we refer to as market clarity, or the informativeness of a good rating, PrGjGR
PrGjN R . This quantity
describes the gain in investors’rational belief that the asset is of high quality, which results from observing a
1 While we focus on loan-backed assets for concreteness, the insights extend to markets for securities backed by corporate debt,
provided that banks involved in debt underwriting and issue of securities act as middlemen between private …rms and investors
(pension funds, college endowment funds, etc), performing the important service of screening and obtaining ratings.
2 Purnanandam (2011) uncovers that banks with greater involvement in secondary markets originated excessively poor-quality
mortgages. Keys, Mukherjee, Seru, and Vig (2010) and Keys, Seru, and Vig (2012) explore loan variation to borrowers with the
credit score around 620, i.e. threshold commonly used in securitization. They …nd that 620+ loans default at the rate 10-25%
above that of the 620- loans. Mian and Su… (2009) provide corroborating evidence from zip-code level data on subprime lending.
Bord and Santos (forthcoming) …nd that loans sold to CLOs at the time of issue are more likely to default.
3 Gorton (2009) points out the severity of asymmetric information due to the loss of information about the quality of the
underlying loans during the securitization process. Beltran, Cordell, and Thomas (2013) argue this type of information asymmetry
was severe enough to cause the collapse of the ABS CDOs market.
2
good rating. In the case of a perfectly accurate rating technology, market clarity is maximal, and the constrained
e¢ cient screening e¤ort and credit allocation may be attained. However, an imperfect rating technology lowers
the informational content of a good rating, both directly, by providing a less accurate signal, and indirectly, by
encouraging the sellers of poor quality assets to try their luck at getting a good rating. As a result, markets
are “confused,” and the premium paid on highly rated assets is too small (i.e. ratings in‡ation) to induce
the constrained e¢ cient level of screening and resource allocation. In general, an intensi…ed use of a rating
technology among the sellers of poor quality assets (i.e. ratings shopping) weakens the screening activity in the
economy. Nonetheless, it is important to emphasize the fact that asset issuers know the quality of their asset
and their rating behavior re‡ects this information. In other words, the fact that sellers of poor quality assets
are less likely to rate their asset helps increase the rating informativeness and encourages screening e¤ort, even
if it still falls short of the e¢ cient level.
The insight provided by the model helps with the interpretation of several puzzling observations of the
expansionary period preceding the 2008 crisis. In the context of our model, both, an increase in collateral values
and a decrease in the rating technology precision, unambiguously reduce the level of screening e¤ort at the stage
of loan origination. Rising collateral values are characteristic of the expansionary period that preceded the 2008
crisis. The declining accuracy of ratings is also relevant because it re‡ects a well-documented growth trend in
asset complexity. Therefore, the model helps us interpret the observation that laxer screening standards were
applied in the period leading up to the crisis. Although the practice of applying looser screening standards in
economic booms is widely documented [see Asea and Blomberg (1998), Berger and Udell (2004), Lown, , and
Morgan (2006), Rajan (1994)], it seems to have been exacerbated by the rise of secondary markets during this
period [see Purnanandam (2011), Keys, Mukherjee, Seru, and Vig (2010), Keys, Seru, and Vig (2012), Bord
and Santos (forthcoming)].
The case of the declining accuracy of rating technology, which re‡ects growing asset complexity, warrants
further comment. The direct implication of a lower rating precision is that it reduces the informativeness of
a good rating. Because mistakes are more likely to happen, this e¤ect is further ampli…ed via the intensi…ed
strategic rating of poor quality assets (ratings shopping). The premium paid on loan baskets with good ratings
falls (i.e. the spread between high yield and low yield assets declines), thereby relaxing the incentives for loan
screening at origination and exacerbating the resource misallocation problem. Therefore, a drop in the accuracy
of ratings, when viewed in the context of our model, helps us interpret additional puzzling features of the precrisis period— an intensi…ed ratings shopping, ratings in‡ation, and a historically low premium paid on highly
rated assets. Gri¢ n and Tang (2011) provide empirical evidence for the intensi…ed ratings shopping and ratings
in‡ation prior to the crisis. The historically low premium on highly rated assets is simply an upshot of ratings
in‡ation. Nonetheless, to provide further evidence of this fact, we compiled a panel, tranche-level, dataset on
asset-backed security deals backed by small business loans (Source: ABSnet).4 The time series of the ratio of
the yield on non-triple-A tranches and tranches with at least one triple-A rating is reported in Figure 1. Clearly,
the yield di¤erential falls dramatically throughout the expansion preceding the …nancial crisis of August 2008,
indicating a drop in the premium paid on highly rated assets. Our explanation for this trend emphasizes the
decline in the informativeness of good ratings.
Finally, we analyze the role of two regulatory policies proposed by the U.S. Securities and Exchange
Commission, namely, mandatory rating and mandatory ratings disclosure. Contrary to conventional wisdom,
we …nd both policies to be counterproductive. Both policies worsen the degree of credit misallocation in the
4 The borrowers’ soft information aquisition that motivates our setup seems to be most relevant in the case of small business
borrowers, which explains our choice of this type of asset backed securities. Small …rms are usually informationally opaque, lacking
audited …nancial statements, thereby making the assessment of their ability to repay non-trivial.
3
economy. Importantly, this result stems from the presence of a feedback e¤ect from …nancial markets on
aggregate activity, which has not been explored in related literature on information production in …nancial
markets as it typically assumes a …xed supply of asset quality.5 One important step towards linking the rating
activity to investment choices is Sangiorgi and Spatt (forthcoming). In that paper, asset issuers themselves
derive information about investment projects from ratings. We focus on a di¤erent part of the economy
where asset issuers have access to an independent screening technology, and ratings are used only to transmit
information to investors.
Under the policy of mandatory rating, rating activity on the part of sellers of low quality assets is intensi…ed,
thereby introducing more confusion in the market, and reducing the informativeness of a good rating. In turn,
the premium paid on assets with good ratings declines, discouraging screening e¤ort at the stage of loan
origination and further exacerbating resource misallocation. Under the policy of mandatory ratings disclosure,
but voluntary rating, sellers of low quality assets are encouraged to rate their assets, as the lack of an observable
rating can no longer be passed o¤ as an undisclosed false rating. Good and bad ratings gain their signalling value
in investors’beliefs. Therefore, strategic rating intensi…es, thereby reducing the informational value of a good
rating, lowering the expected return to screening, and compounding resource misallocation in the economy. This
result is reminiscent of the one derived in Povel, Singh, and Winton (2007), where the mandatory information
disclosure from …rms to investors also generates an unintended feedback e¤ect by generating more fraud.
In light of our analysis, policy design aimed at raising rating accuracy (i.e. reducing asset complexity or
disciplining the behavior of rating agencies) would be much more e¤ective at aligning the equilibrium outcome
with the e¢ cient one. Thus, our work emphasizes the importance of studying the incentive problems of rating
agencies [e.g. Damiano, Li, and Suen (2008), Bolton, Freixas, and Shapiro (2012), Goldstein and Huang (2016),
Cohn, Rajan, and Strobl (2016)].
A closely related study of the real implications of risk shifting is Allen and Gale (2000), in which banks
are endowed with funds but must lend to entrepreneurs to generate returns. While the entrepreneurs make
resource allocation choices, they are protected by limited liability, and thus banks are the ultimate risk bearers.
As a result, entrepreneurs engage in excessive risk taking. Our study is complementary to Allen and Gale
(2000), as we focus on a di¤erent margin. In our model, resource allocation is determined by banks, through
their screening activity. Because banks must sell o¤ their loans to investors, i.e. the ultimate risk bearers, they
engage in suboptimal screening practices.
The rest of the paper is organized as follows. Section 1 describes the model. In Section 2, we analyze
the equilibrium, its existence and uniqueness. In Section 3, we employ the model to gain insight into pre-crisis
empirical trends. In Section 4, we study mandatory rating and mandatory ratings disclosure policies. We
conclude in Section 5. All proofs, unless otherwise stated, appear in the appendix.
2.
Model
We consider an economy with investors, heterogeneous borrowers, and heterogenous banks. We model borrowers
in a very general way so that our setup is applicable to any sector characterized by informationally opaque
borrowers relying on bank …nancing. Some examples would include small businesses, young …rms, and …rst
time home buyers.
5 With
respect to this literature, our paper is most closely related to Skreta and Veldkamp (2009), in the sense that ratings
informativeness is an endogenous object determined by the sellers’ rating and disclosure choices. Skreta and Veldkamp (2009)
must assume that investors are naive (not aware of the possibility of hidden information) in order to avoid unraveling of trade (see
Shavell (1994)). In our model, investors are perfectly rational, and equilibrium exists because banks are in need of investors’funds.
4
Borrowers desire to borrow, investors desire to save, and banks alone have the technology to screen and
identify repaying borrowers. Banks need to raise funds by selling loans to investors.6 But investors do not possess
the information about the underlying quality of traded assets. The importance of studying the macroeconomic
implications of this type of informational asymmetry is discussed in Gorton (2009). Our goal is to examine its
implications for the allocation of loanable funds, i.e. the composition of …nanced borrowers.
The model period can be subdivided into three stages occurring in the following order.
1. Screening of Borrowers. Banks choose whether or not to engage in costly screening of borrowers when
originating loans, taking asset prices as given. Upon origination, loan quality is revealed to the banks.
This stage determines the composition of borrowers.
2. Rating of Assets. Banks choose whether or not to engage in costly rating of their assets, taking asset
prices as given. This stage determines how much information is produced to mitigate the information
asymmetry friction in asset markets.
3. Asset Trade. Banks and investors trade in competitive loan markets, where asset prices are determined.
2.1.
Borrowers
There is a continuum of measure 1 of potential borrowers of unobserved type, each of whom seeks …nancing
in the amount of 1 unit of funds. Potential borrowers are of unobservable type
proportions
0
and 1
0,
respectively. Type
repays W on the loan.
2 fG; Bg; represented in
Assumption 1 We assume that WG > WB :
2.2.
Banks
There is a continuum of measure 1 of pro…t-maximizing banks, heterogeneous in their screening cost k
F [0; 1] ;
which is unobserved to investors. F is continuous and represents the cumulative distribution function of banks’
screening costs. Each bank faces its own pool of potential borrowers of types
2 fG; Bg; in proportions
0
and
1
0 . Banks have the option of using the screening technology at the cost k; which guarantees …nancing of a
good project ( = G). Otherwise, the borrower is chosen at random. We interpret lending to a type borrower
very generally as standing in for extending a large basket of loans that generates W as total proceeds.7
Once the borrower is …nanced, the asset type
is revealed to the bank. This information regarding the
quality of underlying loans is not available to investors. At a cost c, banks can employ a rating technology that
reveals the true asset type with probability r 2 (0:5; 1]. Because banks holding type G assets are more likely to
obtain a good rating on their assets, good ratings will serve as positive signals in secondary loan markets, and
assets with good ratings will sell at a premium.8
Assumption 2 We assume that c < WG
WB :
6 This assumption guarantees the presence of secondary markets for loans, and we do not attempt to explain their emergence
(see Parlour and Plantin (2008)).
7 This would include interest and loan repayments, and collections in case of default.
8 In theory, the problem of information asymmetry between banks and investors could be resolved if the originating banks
retained the most risky junior tranche of their loan basket, thereby sending a credible signal to asset buyers (e.g. DeMarzo (2005)).
In practice, however, retaining a junior tranche does not appear to accomplish this purpose, as it can be combined with shorting
of a senior tranche. In fact, Beltran, Cordell, and Thomas (2013) documents that it is more common for asset issuers to retain
the senior tranches. Nonetheless, we have also worked out a version of our model in which banks are required to hold a positive
fraction of their assets. All of the results go through.
5
Rating an asset in our model should be interpreted as engaging in a costly process that results, with some
positive probability, in the enhancement of the perceived value of the asset. Indeed, the process of getting all
of the existing rating agencies to assign a triple-A rating to a large share of one’s loan basket is a costly process
that involves hiring consultants to obtain better information regarding the rating process of each agency and to
decompose the loan basket into tranches in a way that maximizes the positive outcome. The assumption that
r > 0:5 simply captures the idea that banks with better assets are more likely to succeed in this process.
We also assume that bad ratings are available for free to all banks, which rules out the signalling value
of poor ratings and ensures that only good ratings will be revealed. Note that when more assets get rated in
our model, for a …xed distribution of issued asset types, there are more undisclosed poor ratings. We therefore
will interchangeably refer to the frequency of use of the rating technology as “rating intensity” or “ratings
shopping.”9
Because the screening cost of originating banks is not observed by investors, it follows that asset prices
will be conditioned only on the presence of a good rating. We denote prices on assets with good ratings and
no ratings (or hidden poor ratings) by PGR and PN R ; respectively. Taking these prices as given, banks choose
their screening and rating strategies to maximize their pro…ts.
Rating Strategy
Because asset prices PGR and PN R are conditioned only on the rating signal, all banks holding an asset
of a given type
will make the same decision with respect to employing the rating technology. We allow for a
mixed strategy f ; which refers to the probability of employing the rating technology.
A bank that holds a type B asset chooses to rate it if the expected gain in the asset price strictly exceeds
the cost of using the technology. The gain is positive, PGR
which happens with probability 1
PN R ; only in the case of an inaccurate rating,
r: The asset is not rated if the expected gain falls short of the cost, and the
mixed strategy is possible otherwise. Formally,
fB = 1
fB = 0
(1
r) (PGR
if
fB 2 (0; 1)
PN R ) > c;
:::
< c;
:::
= c:
(1)
We restrict attention to the range of parameter values, derived in Lemma 1, that ensure that it is optimal
to rate high quality assets (fG = 1), i.e. that
r (PGR
PN R ) > c
(2)
holds in equilibrium. From now on, we will assume fG = 1 always holds in equilibrium.
Screening Strategy
We now turn to the screening decision. Each bank, characterized by their screening cost k , chooses whether
or not to screen, taking the asset prices PGR and PN R and the optimal rating strategies fB and fG as given.
Denote by R the expected return from having issued an asset of type . Because the rating technology is
always employed by banks with high quality assets, the expected return from the type G asset is given in (3) :
With probability r, the bank obtains a high rating, reveals it, and sells the asset for PGR . With probability
1
r, this bank obtains a poor rating, hides it, and sells the asset for PN R . The return is reduced by the rating
cost and the unit loan amount. Likewise, the expected return from having the type B asset is given in (4) :
With probability (1
9 This
r) fB , the bank obtains a high rating, reveals it, and sells the asset for PGR . Otherwise,
is broadly consistent with the literature on credit ratings which typically entails more than one rating draw.
6
the bank sells the asset for PN R , which happens both, in the case of the accurate rating draw and the case of
not rating the asset. The return is reduced by the expected rating cost and the unit loan amount. Therefore,
RG
= rPGR + (1
RB
=
(1
r)PN R
c
r)fB PGR + [1
(1
1;
(3)
r)fB ] PN R
fB c
1:
(4)
Banks that choose to screen …nance a high quality loan basket with certainty. The ex-ante expected return
for these banks is given by RG k: Banks that choose not to spend resources on screening draw loan baskets
at random and …nance high quality assets with probability 0 : The ex-ante expected return for these banks
is given by
0 RG
+ (1
0 )RB :
It follows that, in order to maximize the ex-ante expected return from their
investment, a bank of type k should screen whenever
RG
k
0 RG
+ (1
0 )RB ;
(5)
i.e. when the screening cost is su¢ ciently small. The marginal screener is the bank of type
k = (1
The more productive banks (k
0 )(RG
RB ):
(6)
k) screen and hence introduce only high quality borrowers into the economy.
The less productive banks forgo the screening process and hence introduce high quality borrowers into the
economy with probability
0.
Recall that the distribution of banks is described by F , which implies that the
marginal screener determines the measure of high quality borrowers in the economy:
= F (k) + (1
2.3.
F (k))
0:
(7)
Investors
Investors save by purchasing loan-backed assets in competitive markets. Because we assumed that investors
have no information regarding the quality of the underlying loans or screening productivity of loan originators,
investors’ beliefs regarding asset quality are conditioned only on the observed ratings. We denote investors’
beliefs regarding asset quality by PrGjGR and PrGjN R : Our assumption of competitive markets implies that
investors make zero pro…ts, i.e. that equilibrium prices on assets with good ratings and no ratings re‡ect their
expected returns, as perceived by investors:
PGR
=
WG PrGjGR + WB [1
PrGjGR ] = PrGjGR W + WB ;
(8)
PN R
=
WG PrGjN R + WB [1
PrGjN R ] = PrGjN R W + WB ;
(9)
where
W = WG
WB .
2.4.
Model Equilibrium
We have thus far described the banks’optimal screening and rating decisions for given asset prices. We have
also discussed how the asset prices are related to investors’ beliefs regarding asset quality. To complete our
7
de…nition of equilibrium, we also require that investors’beliefs are consistent with the equilibrium outcomes:
PrGjGR
=
PrGjN R
=
r
;
r + (1
)fB (1 r)
(1 r)
:
(1 r) + (1
)[(1 fB ) + fB r]
(10)
(11)
PrGjGR corresponds to the actual fraction of high quality assets among the highly rated assets, i.e., among the
high quality assets with accurate ratings and the poor quality assets with inaccurate ratings. Similarly, PrGjN R
corresponds to the actual fraction of high quality assets among all unrated assets. We will refer to the belief
di¤erential
Pr := PrGjGR
PrGjN R
as the (good) rating informativeness, or market clarity, precisely because it captures the increment in the
equilibrium probability that the asset is of high quality implied by an observation of a high rating. This
endogenous quantity will play a critical role in generating all of our results. We can immediately see that the
rating informativeness determines the premium paid on highly rated assets as well as the rating intensity.
Definition. An equilibrium is given by the marginal screener k , the rating strategies fG = 1 and fB , the
measure of good borrowers in the economy
; investors’ beliefs regarding asset quality conditional on the
observed rating fPr GjGR ; Pr GjN R g and asset prices conditional on the observed rating fPGR ; PN R g satisfying
the following conditions:
1. Given the asset prices fPGR ; PN R g, banks of type k
k …nd it optimal to screen their borrowers (i.e.
the screening condition (5) is satis…ed for these banks), while banks of type k > k …nd it optimal to select
their borrowers at random (i.e. the screening condition (5) does not hold for these banks).
2. Given the asset prices fPGR ; PN R g, banks that issued type
assets …nd it optimal to rate according to f
(i.e. the rating optimality conditions (1) and (2) are satis…ed).
3. Given investors’beliefs fPr GjGR ; Pr GjN R g, asset prices fPGR ; PN R g re‡ect expected returns: conditions
(8) and (9) hold.
4. Investors’ beliefs fPr GjGR ; Pr GjN R g are consistent with the equilibrium outcomes: (10) and (11) hold.
5. The measure of resource allocation in the economy
strategies, summarized by k :
= F (k ) + (1
2 (0; 1) is determined by the optimal screening
F (k ))
0:
(12)
To the extent that borrower repayment W is related to the borrower productivity in whatever entrepreneurial activity the borrower seeks to …nance, the equilibrium object
re‡ects the average productivity in
the sector that relies on bank …nancing. In section 3.4, we derive the e¢ cient level of screening k ef : In the
spirit of Boyd and Prescott (1986) and Holmstrom and Tirole (1997), banks perform an important service of
screening of borrowers, thereby directly a¤ecting aggregate productivity through the credit allocation margin
summarized by
. Our model allows us to analyze how the presence of information asymmetry in markets
where loan-backed assets are traded a¤ects the intermediaries’incentives to screen.
8
3.
Equilibrium Characterization
We solve for the equilibrium quantities and prices as follows. We …rst characterize the optimal rating strategy
fB as a function of ; which ensures that Conditions 2-4 in the de…nition above hold. This is done in Lemmas
1 and 1. Once the equilibrium relationship fB ( ) is derived, it is straightforward to obtain beliefs and prices as
functions of
as well. Given these prices, the marginal screener k is determined in accordance with Condition
1 in the de…nition. Finally, the equilibrium resource allocation
is found as a …xed point of (12) provided in
Condition 5. Existence and uniqueness of the equilibrium are proved in Proposition 1.
3.1.
Rating Strategy
We now proceed to characterize the rating strategy fB ; for a …xed . The optimal rating strategy fB ; described in
(1) ; depends on asset prices. In turn, asset prices depend on investors’beliefs through the zero pro…t conditions
(8) and (9) ; and beliefs depend on and fB through the consistency conditions (8) and (9) : Therefore, by
substituting consistent beliefs into prices and prices into the rating optimality condition, we can determine the
equilibrium dependence of the rating strategies fB on the fraction of good borrowers in the economy .
It is …rst helpful to describe how the rating accuracy, r, and the cost of screening relative to the repayment
di¤erential,
c
;
W
c~
(13)
a¤ect the rating decision. The lemma below characterizes the optimal rating strategy in the parameter space
of r and c~ (for a …xed ).
Lemma 1 Rating strategies fB and fG in the space of c~ and r (for a …xed ).
For a given measure of resource allocation, , the rating strategy fB can be summarized as follows:
fB =
8
>
>
<
>
>
:
1
fBmix 2 (0; 1)
0
(1
)(1 r)(2r 1)
[r
(2r 1)][ (2r 1)+(1 r)]
(1 r)(1
)
1 r
c~
if
[r
(1
)(1 r)(2r 1)
(2r 1)][ (2r 1)+(1 r)]
(1 r)(1
)
1 r
where the mixed strategy is given by fBmix =
c~(1 2 r)+ (1 r)
<
c~ <
(14)
c~;
p
(~
c+ )2
2~
c(1 r)(1
2 r(2
r) (6 4r)~
cr
)
:10
Moreover,
fG = 1
if c~ <
r (1
)
:
1 r
(15)
Proof. See the appendix.
Figure 2 helps visualize this result. It illustrates the optimal rating choice for an arbitrarily chosen
Recall that the asset is rated, fB > 0; whenever (1
r) Pr
= 0:5:
c~. Higher levels of c~ clearly weaken the incentive
to rate poor quality assets. However, the e¤ect of r depends on its level. On one hand, raising r increases
rating accuracy, making it less likely that a poor quality asset obtains a lucky high rating and collects the
premium. This e¤ect discourages rating of poor quality assets. On the other hand, raising r increases the
rating informativeness
Pr and the premium paid on highly rated assets. This e¤ect tends to encourage rating
activity. The latter e¤ect dominates for low r: When r is at its lowest, near 0:5, high ratings are largely
uninformative, which implies a low premium. As a result, poor quality assets are seldom rated, despite the high
likelihood of obtaining a high rating by mistake. As r increases, so does the premium paid on high ratings,
1 0 It
is found by solving (1
r) (PGR
PN R ) = c after substituting for prices and beliefs.
9
which encourages rating activity. Conversely, the former e¤ect dominates for large values of r. When r is near
1, the poor quality asset has no chance of obtaining a high rating and its issuer is discouraged from rating it
regardless of the premium paid on high ratings. This intuition applies to the entire range of the parameter
space. Note, however, that when c~ is su¢ ciently high, fB never increases above 1 because c~ is too high.
Our focus on the parameter space that ensures that fG = 1 in equilibrium implies that the range of
admissible
is between 0 and
:=
r c~
;
r(1 c~)
found from (15). Corollary 1 formalizes the general characterization of the equilibrium relationship fB ( ) :
[0; ] ! [0; 1] : We show that fB is 0 when
= 0. It is increasing for low levels of
and may or may not reach
the level of 1 before it becomes a decreasing function of : For su¢ ciently high levels of ; the rating strategy
fB reaches the level of 0 and stays at that level as
increases further. To understand this shape, recall the
rating optimality condition (1) which reveals that the return to rating the issued asset is disciplined by the
premium paid on good ratings (PGR
easy to see that
PN R ), which, in turn, depends on the rating informativeness
Pr is a single-peaked function of , increasing for small values of
Pr. It is
and decreasing for high
values of . The reason for this is as follows. When is small, there are very few good assets in the economy,
so a good rating does little to change investors’beliefs regarding the asset quality. The same is true for high
levels of
when all traded assets are predominantly of high quality. For intermediate values of
good rating carries a lot of informational content. Therefore, it is for intermediate values of
though, a
that investors
are willing to pay the highest premium on good ratings and banks are most likely to try their luck at obtaining
a high rating.
Corollary 1 Rating strategy fB ( ) : [0; ] ! [0; 1] as a function of resource allocation,
If c~ > (1
r) (2r
1) ; then
fB ( ) =
with fB increasing for
2 (0;
de…ned as
max )
max
If instead c~
(1
r) (2r
=
with fB increasing for
1
=
2
8
>
<
0
fBmix
>
:
= 0;
( ) 2 (0; 1) if
2 (0;
0
and decreasing for
1
2 (1
(1
2
r)
2
2(
max ;
c~2
r)
c~ (2r
1) (1
2[
r)
;
3 ),
=
3
3) ;
3 ; );
where the constants
1 r
1 r
max
c~
:
r~
c
and
3
are
(16)
1) ; then
fB =
1
:
2 (0;
1
2
s
1)
8
>
0
>
>
>
mix
>
>
< fB ( ) 2 (0; 1)
= 0;
if
1
>
>
>
fBmix ( ) 2 (0; 1)
>
>
>
:
0
and decreasing for
10
2[
2(
2(
(1 r) (2r 1) c~
;
(2r 1) (~
c + 1 2~
cr r)
2 (0;
2
2;
3) ;
1) ;
1;
2 ];
2;
3) ;
3;
where the constants
1 1
= +
2 2
s
2
are de…ned as
(1 r) (2r 1) c~
:
(2r 1) (~
c + 1 2~
cr r)
(17)
1
and
Proof. Notice that
2
1
<
<
max
2
<
3
< : The proof follows from Lemma 1 after recognizing that
denote the lower and higher solutions of
fBmix
= 1, that
= 0; and that
max .
of 1. If instead c~
1) ; then fBmix will exceed 1 on the interval (
r) (2r
r) (2r
solves
U-shaped function maximized at
(1
If c~ > (1
3
fBmix
fBmix
1) ; the rating strategy
1;
fBmix
1
and
is an inverted
never reaches the level
2 ) ; implying rating with
certainty in that range.
3.2.
Screening Strategies and Resource Allocation
Now that the equilibrium relationship fB ( ) is characterized in Corollary 1, it is straightforward to obtain the
equilibrium dependence of investors’beliefs and prices on . Given these prices, the marginal screener k ( ) is
determined according to (6):
k ( ) = (1
0)
R( );
(18)
where the return di¤erential is given by
R( )
:
= RG ( )
RB ( )
=
(PGR ( )
PN R ( )) [r
=
W Pr ( ) [r
(1
(1
r) fB ( )]
r) fB ( )]
(1
(1
fB ( )) c
fB ( )) c:
(19)
An important insight emerges. It is the price di¤erential on assets with good ratings and assets with no ratings
that disciplines the banks’screening e¤ort at the stage of loan origination. In turn, as seen from the above, this
premium is proportional to the rating informativeness
Pr ( ) =
r + (1
r
)fB ( ) (1
r)
Pr( ) :
(1
r) + (1
(1 r)
;
)[(1 fB ( )) + fB ( ) r]
(20)
obtained by substituting for the beliefs from (10) and (11). Rating informativeness is high when the rating
technology is accurate and poor quality assets are not rated. In that case, a good rating implies a large gain in
the perceived quality of the asset, investors pay a large premium on assets with high ratings, and more banks
screen their borrowers. In the next subsection, we will show that as long as the rating technology is prone to
error (r < 1) ; rating intensity confuses the market and rating informativeness is insu¢ cient to imply the …rst
best allocation. However, as long as the rating technology is prone to error, good ratings will not be su¢ ciently
informative to induce the optimal level of screening e¤ort.
Nonetheless, it is important to emphasize that the fact that fB ( ) < 1 for a large range of admissible
is important in generating an additional wedge between PrGjGR and PrGjN R : The wedge already exists even in
the case of fB ( ) = 1 because r > 0:5, but it is larger when fB ( ) < 1: In other words, asset issuers know
the quality of their asset and this information is re‡ected in their rating behavior. Less frequent rating of
poor quality assets helps increase the rating informativeness and encourages the screening e¤ort. We discuss
in Section 4 that policies that have been o¢ cially proposed to combat the informational asymmetry friction
in asset markets are likely to be counterproductive exactly because they eliminate this endogenous wedge that
strengthens the informativeness of the good rating.
Substituting the marginal screener expression (18) into the …nal equilibrium condition (12) ; it becomes
clear that the resource allocation in the economy
can be obtained as a …xed point of
= F (k ( )) + 1
11
F (k ( )
0:
(21)
To prove existence and uniqueness of equilibrium, it su¢ ces to show that there is a unique solution
to the
above equation. The proof is formalized in Proposition 1.
The remaining equilibrium quantities and prices have already been characterized as functions of : The
equilibrium rating strategy fB can be determined from (14) : Using this information, investors’beliefs fPrGjGR ;
PrGjN R g are computed from consistency conditions (10) and (11) : Given those, asset prices are computed from
the zero pro…t conditions (8) and (9) :
Proposition 1 Existence and uniqueness of equilibrium
h
i
)
Denote the right hand side of the equilibrium condition (12) by H : 0; r(1
![
1 r
H( ) := F (k( )) + 1
Assume
0
< : If parameter values satisfy c~
2
2
0)
f (1
(2r 1)
W
r (1 r)
s
(1
r) (2r
F (k ( )
0 ; 1] ;
0:
(22)
1) ; then also assume
1 r c~= (2r 1)
=
1 r c~ (2r 1)
1
1
r
r
c~ (2r
2
< 1;
1)
(23)
where f := sup F 0 (k).
k2[0;1]
Then H 0 ( )
1 and there exists a unique equilibrium.
Proof. See the appendix.
Note that the condition (23) imposed to ensure uniqueness is quite weak. In fact, this condition is not
needed for parameterizations that imply that the equilibrium measure of high quality assets is over a half.11
3.3.
Ine¢ ciency of the Decentralized Equilibrium Outcomes
It is instructive to compare outcomes in the decentralized economy to the constrained e¢ cient outcomes. In
order to de…ne the constrained e¢ cient allocation, however, we need to specify a few more details about the
production structure of the economy. For simplicity, we assume that the rating cost c is simply a transfer from
banks to rating agencies. We also assume that there are no additional costs associated with production of
projects underlying the loan baskets, apart from the unit of funds extended, given in units of the …nal good.
Suppose further that extending a loan basket of type G results in output production in the amount of YG ,
whereas extending a loan basket of type B results in output production in the amount of YB , where YG > YB .
Finally, we note that the relationship between the repayment amounts and the actual output levels will satisfy
WG
WB
YG
YB .12
It follows that the total output of the economy is a function of the level of screening activity k:
Y = F (k)YG + (1
F (k))(
0 YG + (1
0 )YB )
Z
k
kdF (k)
1:
(24)
0
The …rst two terms give the output of the …nal good produced by projects underlying the loan baskets; the last
two terms represent inputs involved in screening and production of projects. With a greater k; more resources
are used up in screening, but the unit of funds employed in production yields more output.
1 1 We
show that H ( ) is weakly decreasing for 2 (0:5; ) without any restriction on parameter values.
relationship is consistent with either debt or equity …nancing contracts. Modeling the bank contracts in more detail is
outside the scope of this paper and irrelevant for our analysis.
1 2 This
12
The constrained e¢ cient allocation is given by the screening activity k that maximizes Y . The social
marginal gain of screening by any bank is (1
probability
0
0 )(YG
YB ), because the good borrowers are …nanced with
even if the bank chooses not to incur the screening cost. The marginal cost for a given bank is k:
Therefore, it is socially optimal for a bank with the screening cost k to screen whenever (1
0 )(YG
YB ) > k:
The most productive bank faces the screening cost k = 0; therefore, it is always e¢ cient for this bank to screen.
The least productive bank faces the screening cost of 1: If (1
0 )(YG
YB ) > 1; then even the least productive
bank should screen, and so should the rest of the banks. Otherwise, there exists a cuto¤ marginal screener
k ef 2 (0; 1) satisfying k ef = (1
0 )(YG
YB ): Formally, the socially e¢ cient marginal screener is given by
k ef = minf(1
0 )(YG
YB ); 1g
(25)
and the implied socially e¢ cient measure of resource allocation is given by
ef
= F (k ef ) + 1
F (k ef )
0:
(26)
In the decentralized economy, k < 1, i.e. it is never the case that all banks screen. If this were the case,
consistent beliefs would imply that all assets are of high quality and all assets would trade for WG . This, in
turn, would imply that no individual bank would have an incentive to engage in screening, which leads to a
contradiction. Even if k ef < 1, the level of screening in the decentralized economy falls short of the socially
e¢ cient level, as long as r < 1. The following proposition formalizes this result.
Proposition 2 Decentralized Outcomes vs. Constrained E¢ cient Outcomes
Whenever r < 1, the equilibrium level of screening activity is less than e¢ cient and resources are misallocated:
k < k ef and
Proof. First consider the case where (1
Suppose not. If k = 1 then
0 )(YG
<
ef
:
1 and so k ef = 1: We want to show that k < k ef = 1:
YB )
= 1; and it follows that PGR (
) = PN R (
) = WG : This, in turn, implies that
no bank will choose to engage in costly screening, i.e. k = 0. We arrive at a contradiction.
Now consider the case where (1
0 )(YG
YB ) < 1 and so k ef = (1
0 )(YG
YB ): Drawing on equilibrium
conditions (18) and (19), we have
k
=
(1
0)
=
(1
0) (
< (1
0)
R( )
W Pr [r
(1
W r Pr < (1
where the inequalities are due to r < 1 and
r)fB ]
0)
(1
fB )c)
0 )(YG
YB ) = k ef ;
Pr < 1 (which is also implied by r < 1):
Recognizing that H ( ) increases in k, the result that
3.4.
W
(1
<
ef
follows immediately from (21) and (26).
Macroeconomic E¤ects of Increasing Asset Complexity and Collateral Values
Rising collateral values ( W ") are characteristic of the expansionary period that preceded the 2008 crisis. The
declining accuracy of ratings (r #) is also highly relevant and will receive our most attention, because it re‡ects
a well-documented trend in growing asset complexity. In this section, we show that both of these changes
exacerbate the resource misallocation in the economy and explore their additional implications for pre-crisis
observations. We also investigate the e¤ects of an improvement in the pool of projects/borrowers ( 0 ") ; which
13
would re‡ect the expansionary rise in the average productivity of borrowers.
Recall our measure of rating informativeness (or market clarity), given in (20) : Market clarity decreases
in rating intensity, i.e.
@ Pr
@fB
< 0: Intuitively, as more poor quality assets are rated, the probability that a
good rating indicates a good asset declines. It is also easy to see that rating informativeness increases in r;
i.e.
@ Pr
@r
> 0: Intuitively, a more precise rating technology increases the gain in the perceived quality of highly
rated assets.
Market clarity is critical for the screening decision of individual banks, and therefore for the measure of
resource allocation in the economy. Ceteris paribus, an increase in the market clarity leads to an increase in
the screening activity of banks.
We proceed to derive the partial e¤ects of various parameters on the equilibrium level of resource allocation
. It is helpful to …rst recall that, in any equilibrium with fB (
are indi¤erent between rating and not rating their asset:
Pr =
c
W (1
r)
) 2 (0; 1); the issuers of poor quality assets
:
(27)
It follows immediately from the above equality that (i) rating informativeness decreases in
W and (ii) rating
informativeness increases in the cost of screening c and rating accuracy r: The intuition for these results invokes
the strategic rating behavior on the part of banks that …nanced poor quality loan baskets. If, for example,
W increases, the mixed strategy will remain viable only if fB also increases, which would work to reduce the
rating informativeness ( Pr). Therefore, in any equilibrium where the banks with low quality assets play a
mixed strategy, a change in the model parameter values, whether it is c;
W or r, has an important indirect
e¤ect on the rating intensity fB and, therefore, on rating informativeness through condition (27). This indirect
e¤ect, then, has important consequences for the screening activity at the stage of loan origination. We proceed
to illustrate that this strategic rating component plays a critical role in the comparative statics analysis.
We begin by analyzing the impact of a change in
decline in
W . We interpret the rise in collateral values as a
W; because more value is recovered from defaulting borrowers in the case of higher collateral prices.
For example, the housing boom preceding the 2008 …nancial crisis would be re‡ected in our model as a decline
in
W . Likewise, strengthening of the relationship between the performance of the projects underlying the
disbursement of funds and performance of the …nancial assets would also mean lower frequency of default, or
higher collections in case of default, which would be re‡ected in our model as a decline in
W . To understand
this interpretation, note that in many environments, a switch from debt …nancing towards equity …nancing
increases productivity of the underlying projects. Our model abstracts from the details of …nancial contracts at
the stage of asset issue, therefore any development in these markets would simply be re‡ected in the repayment
levels WB and WG :
Intuitively, an increase in the repayment di¤erential
W should increase the measure of banks that screen
as it directly increases the premium paid on highly rated assets, which raises the return di¤erential
R. This
is indeed true if banks with low quality assets choose a pure rating strategy. In other words, for small changes
in
W , the rating behavior remains una¤ected, so there are no additional e¤ects operating. In the case of a
mixed rating strategy though, fB will intensify and rating informativeness will decline, in line of our discussion
of equation (27). We …nd that the positive direct e¤ect of
W on screening is exactly o¤set by an increase in
the intensity of rating the poor quality assets. Lemma 2 formalizes these results.
Our model is consistent with the view that the rise in collateral values in mortgage or business loan markets
( W #) may have worked to weaken the screening incentives of the loan originators, thereby worsening the
average quality of the …nanced borrowers.
14
Lemma 2 Macroeconomic e¤ ects of the change in
W
The following e¤ ects on screening e¤ ort k and measure of resource allocation
a) If fB (
) 2 (0; 1); then k and
are independent of
b) If fB (
) 2 f0; 1g; then k and
are strictly increasing in
hold in equilibrium:
W:
W:
Proof. See the appendix.
Rising asset complexity was one of the main features characterizing the rise in markets for securitized
products – a widespread phenomenon that took place prior to the 2008 …nancial crisis. In the context of our
model, rising asset complexity is re‡ected in the decline of a rating precision (r #). New important insight
emerges with respect to the e¤ect of rising asset complexity on screening, credit allocation, ratings shopping
and ratings in‡ation.
Lemma 3 formalizes the e¤ect of rating accuracy on screening e¤ort and the measure of resource allocation.
An increase in the rating precision directly raises
R (given in (19)) by increasing the probability that high
quality assets will receive a good rating and sell at a premium, and by decreasing the probability that poor
quality assets will receive a good rating in error and sell at a premium. This clearly encourages banks to expend
resources on screening. Moreover, the premium itself also rises, further reinforcing the incentive to screen. The
reason for this is that the rating informativeness increases with r: When r is near 0:5, good ratings are largely
uninformative because the poor quality assets can obtain them just as well as the high quality ones. The
opposite is true when r is near 1.
Lemma 3 Macroeconomic e¤ ects of the change in rating accuracy r
In equilibrium, screening e¤ ort k and measure of resource allocation
are strictly increasing in r:
Proof. See the appendix.
Figure 3 illustrates the numerical experiment of lowering the rating precision. As we move along the
X-axis, r falls to represent the increase in asset complexity. We observe that the strategic rating on the part
of lemon holders intensi…es (panel b). For a su¢ ciently high rating accuracy, poor quality assets are never
rated, and for a low enough level of rating precision, everyone engages in rating.13 Rating informativeness,
depicted in panel f, decreases through the direct e¤ect of ratings becoming more prone to error, which reduces
the signaling value of a good rating. But in the region where a mixed strategy is played, rating informativeness
and, therefore the premium paid on highly rated assets, decline faster (panel f). In this region, investors’beliefs
regarding the relative quality of highly rated assets deteriorate due to both— the direct e¤ect of less accurate
ratings and the indirect e¤ect of the intensi…ed rating of poor quality assets. In the same region, the fraction
of banks screening and the measure of good assets in these loan markets decline most rapidly (panels a and c).
This is not surprising, because the screening decision depends on the premium paid on highly rated assets and
on the probability of obtaining a good rating, both of which decline.
Perhaps most interestingly, we observe the possibility that the measure of highly rated assets actually
increases concurrently with the declining actual measure of high quality assets in the economy (panels a and
e). This happens in the region of the mixed strategy. The reason for this regularity is the intensi…ed rating
behavior on the part of banks holding poor quality assets. As the rating accuracy drops and more of the asset
issuers try out their luck at getting a good rating in order to sell at a premium, the measure of highly rated
assets rises.
1 3 This e¤ect is not unambiguous. In fact, for a low enough rating precision, r near 0:5, a good rating would be largely
uninformative and would induce a low premium, which would reduce the incentive to rate.
15
Therefore, both the decline in the accuracy of ratings and rising collateral values (falling r and
W ) help
rationalize several puzzling phenomena observed prior to the …nancial crisis:
(1) Laxer screening standards, as identi…ed by the empirical papers cited in the Introduction. This is
re‡ected in our model as less screening and a greater degree of resource misallocation resulting from both
falling r and
W:
(2) An intensi…ed use of ratings, or shopping for high ratings. The model produces this feature as a result
of a declining r (panel b or Figure 3).
(3) The rise in default probability conditional on high rating. The model produces this fact as a result of
a declining r (panel d of Figure 3).
(4) Historically low spreads between high yield (low rated) and investment grade (highly rated) securities.
The model produces the fall in this spread a result of a declining r (panel f of Figure 3) and suggests that it
is possible for the fraction of highly rated assets to rise despite the worsening degree of credit misallocation
allocation (panel e of Figure 3).
We wish to emphasize that neither of the above facts can be studied in a model of asset markets that
treats the asset quality distribution as …xed. It is imperative to model the interaction of rating intensity in
markets between asset issuers and investors with the incentives to screen the underlying projects at the stage
of asset issue. By linking the interaction of information production in asset markets to the decision to screen,
our model o¤ers new insight into several puzzling empirical facts that characterized the time period leading up
to the …nancial crisis.
To complete our comparative analysis, it remains to consider the quality of the pool of borrowers (
0)
and
the cost of rating c. Perhaps the most surprising result in this section is that an improvement in the pool of
borrowers, 0 , may actually worsen the degree of resource misallocation in the economy, i.e. imply a lower :
This is formalized in Lemma 4. An increase in 0 clearly has a direct positive e¤ect on . Ceteris paribus, it
can only increase the proportion of good borrowers in the economy. However, an increase in
0
also weakens
the incentive to screen as it implies a better ex-ante distribution of borrowers and makes the option of choosing
the loan basket at random more attractive. Condition (28) compares the size of these two e¤ects. The positive
e¤ect amounts to 1
F (k), as it operates only through measure 1
random. The negative e¤ect equals (1
0 )Fk (k)
F (k) of banks that select their borrowers at
R. It is given by the measure of banks that stop screening
Fk (k) R multiplied by the loss in the probability of issuing a high quality asset associated with the decision
to pick at random (1
0) :
Lemma 4 Macroeconomic e¤ ects of the change in the pool of borrowers (
The following e¤ ects on screening e¤ ort k and measure of resource allocation
a) k is strictly decreasing in
b)
is weakly increasing in
0)
hold in equilibrium:
0:
0
if and only if
1
F (k (
))
(1
0 )Fk (k(
)) R:
(28)
Proof. See the appendix.
Lemma 5 formalizes the e¤ects of the rating cost. Because high quality assets are rated with certainty,
an increase in the rating cost c will raise the expected rating cost of a high quality loan basket relative to
that of a poor quality loan basket as long as fB < 1. This e¤ect discourages screening, and this is the only
e¤ect operating if fB = 0: Of course, the incentives to screen are una¤ected if fB = 1, because the rating costs
increase proportionately. If a mixed rating strategy is played, then there is an additional e¤ect of an increase
16
in c that dominates. A greater c reduces the incentive to rate poor quality assets, thereby raising the signaling
value of high ratings. This raises the premium paid on highly rated assets, thereby strengthening the incentive
to screen.
Lemma 5 Macroeconomic e¤ ects of the change in the rating cost (c)
The following e¤ ects on screening e¤ ort k and measure of resource allocation
a) If fB (
) = 0; then k and
b) If fB (
) 2 (0; 1); then k and
c) If fB (
) = 1; then k and
hold in equilibrium:
are strictly decreasing in c:
are strictly increasing with c.
are independent of c.
Proof. See the appendix.
4.
Policy Experiments
The previous section showed that the strategic rating component plays a critical role for the model results.
In this section, we analyze mandatory rating and mandatory ratings disclosure (but voluntary rating) as two
policies that may potentially align the decentralized equilibrium allocation with the e¢ cient one. We …nd that,
contrary to conventional wisdom, both policies worsen the degree of resource misallocation.
4.1.
Mandatory Rating
Consider a policy that dictates that all assets must be rated. At …rst glance, this policy may appear appealing
because the rating technology is indeed informative (r > 0:5) and its employment will produce more information
in the asset market. We show that this policy is at best ine¤ective. It makes no di¤erence in the case where all
assets are already rated in the equilibrium. In all other cases, the policy unambiguously worsens the degree of
resource misallocation.
Under mandatory rating, the possibility of hiding the obtained ratings is irrelevant. Because it is perfectly
known that everyone is required to obtain a rating, a lack of a rating will signal a hidden poor rating. Therefore,
an introduction of the mandatory rating policy into our model is equivalent to setting fB = fG = 1.
We show that by intensifying the rating activity on the part of sellers of low quality assets, this policy
introduces more confusion in the asset market, reducing informativeness of a good rating and therefore dampening the premium paid on highly rated assets. We emphasize that the key intuition is in the fact that the
premium paid on highly rated assets (PGR
PBR ) declines in fB : In turn, the premium paid on assets with
good ratings is what disciplines screening e¤ort at the stage of loan origination. As this premium declines as a
result of policy introduction, banks are discouraged from expending their resources on screening, and the degree
of resource misallocation worsens.
Denoting the measure of good assets in the economy under mandatory rating by
results in the proposition below.
Proposition 3 Macroeconomic Implications of the Mandatory Rating Policy
Under the mandatory rating policy, the resource misallocation worsens:
:
mr
17
mr
, we formalize our
The following comparative statics results hold:
@
mr
@r
@
@
Finally,
mr
0
> 0;
@
mr
@c
= 0;
@ mr
> 0:
@ W
> 0 if and only if inequality (28) holds.
Proof. To show the …rst result, it su¢ ces to show that H ( ) ; de…ned in (22) ; is lower under the mandatory
rating policy. This would imply that the …xed point of
= F (k( )) + (1
Recognizing that H ( ) increases in k ( ) and therefore in
Rmr ( )
F (k( )))
0
is lower under this policy.
R; it su¢ ces to show that
R( )
(29)
for admissible : This result follows immediately from the de…nition of
R
=
(r
(1
r) fB ) (PGR
PBR ) c (1 fB )
r
r + (1
)fB (1 r)
=
(r
(1
r) fB )
(1
(1
r) + (1
r)
)(1 fB (1
r))
c (1
fB )
after employing the expression of fB ( ) given in Corollary 1. Condition (29) is satis…ed with strict inequality
in the range of
where fB ( ) < 1 in the benchmark model.
The comparative statics results follow from Lemmas 2-5 after substituting for fB = 1.
This policy experiment highlights the importance of the distinct rating activity of high quality and low
quality assets. The fact that poor quality asset issuers know they have a lemon and this translates into less
frequent rating shopping on their part. This is important because it helps raise the ratings informativeness,
and therefore, discipline the screening activity. Our results suggest that e¤ective policy design should take
this important channel into account. In light of our analysis, policy design aimed at raising r (i.e. reducing
asset complexity or disciplining the behavior of rating agencies) would be much more e¤ective at aligning the
equilibrium outcome with the e¢ cient one.
4.2.
Mandatory Ratings Disclosure
We now consider a policy that dictates that all ratings must be disclosed, although leaves the banks to freely
make their rating choices. This policy also appears potentially useful as it simply suggests to disclose the
information that has already been obtained. However, we …nd that, for a broad range of parameter values, this
policy intensi…es the rating activity of bad assets, weakens the incentive to screen and exacerbates the resource
misallocation problem in the economy.
With the introduction of mandatory ratings disclosure, investors can di¤erentiate between assets with good
ratings, assets with bad ratings, and unrated assets, all of which will trade at potentially distinct prices that
we denote by PGR , PBR ; and PN R , respectively. Therefore, we must adjust several de…nitions in the model
economy and review the rating strategies.
A bank with a high quality asset chooses to employ the rating technology (fG = 1) if and only if
rPGR + (1
r)PBR
c > PN R :
As in the benchmark model, we consider only the range of parameter values implying that fG = 1.
18
(30)
Similarly, a bank with a poor quality asset employs the rating technology (fB = 1) if and only if
(1
r) PGR + rPBR
c > PN R ;
(31)
it is indi¤erent towards rating if the above expression holds with equality and prefers to opt out of rating
otherwise.
The expected returns from having issued assets of type
RG
=
rPGR + (1
RB
=
fB [(1
r)PBR
c
r)PGR + rPBR
2 fB; Gg are then given by
1;
c] + (1
(32)
fB )PN R
1:
(33)
Our assumption of competitive markets implies that investors make zero pro…ts, i.e. that equilibrium
prices on assets with good ratings, bad ratings and no ratings re‡ect their expected returns, as perceived by
investors:
PGR
=
WG PrGjGR + WB [1
PrGjGR ] =
W PrGjGR + WB ;
(34)
PBR
=
WG PrGjBR + WB [1
PrGjBR ] =
W PrGjBR + WB ;
(35)
PN R
=
WG PrGjN R + WB [1
PrGjN R ] =
W PrGjN R + WB :
(36)
Finally, we require that investors’beliefs are consistent with the equilibrium outcomes:
PrGjGR
=
PrGjBR
=
PrGjN R
=
r
;
)fB (1 r)
(1 r)
;
r) + (1
)fB r
r + (1
(1
0:
(37)
(38)
(39)
The last belief is zero because high quality assets are always rated (fG = 1) and disclosed.
Other than the fact that three types of assets are now traded in asset markets, the equilibrium in the
economy under the mandatory ratings disclosure policy is de…ned as in the benchmark model. It is given by
the marginal screener k , the rating strategies fG = 1 and fB , the measure of good borrowers in the economy
; investors’beliefs regarding asset quality conditional on the observed rating fPrGjGR ; PrGjBR ; PrGjN R g and
asset prices fPGR ; PBR ; PN R g such that:
1. Given the asset prices, banks of type k
k …nd it optimal to screen their borrowers, i.e. the screening
condition (5) is satis…ed when we substitute for RG and RB de…ned in (33) and (32), while the opposite
is true for banks of type k > k .
2. Given the asset prices, banks that issued type
assets …nd it optimal to rate according to f (i.e. the
rating optimality conditions (30) and (31) are satis…ed).
3. Given investors’beliefs, asset prices re‡ect expected returns: conditions (34) - (36) hold.
4. Investors’beliefs fPrGjGR ; PrGjN R g are consistent with the equilibrium outcomes: (37) - (39) hold.
5. The measure of resource allocation in the economy
strategies, summarized by k :
= F (k ) + (1
19
2 (0; 1) is determined by the optimal screening
F (k ))
0:
First of all, we …nd that under mandatory disclosure, it is never the case that fB = 0: The intuition
is as follows. If fB = 0 and fG > 0, then the disclosure of any rating, whether it is good or bad, would
imply with certainty that the asset is of high quality. The equilibrium asset prices generated by such beliefs
would incentivize poor asset issuers to rate their assets, which contradicts the assumption of fB = 0: Lemma 6
characterizes the rating strategy fB for a given measure of resource allocation :
Lemma 6 mandatory ratings disclosure: Rating strategy fB (for a …xed ).
For a given measure of resource allocation,
2 (0; 1], the rating strategy fB can be summarized as follows:
fG = 1 and
fB =
(
1
fBmix 2 (0; 1)
c~ <
if
r(1 r)
)(1 r)][ (1 r)+(1
[ r+(1
where the mixed strategy is given by fBmix =
)r]
p
(1+2~
c)(r r 2 ) c~+
r(1 r)
)(1 r)][ (1 r)+(1
[ r+(1
)r]
c~ < 1;
10~
cr 2 (16~
c+2)r 3 +(8~
c+1)r 4 +4~
c2 (r 2 r)+(~
c r)2 14
.
2~
cr(1
)(1 r)
Proof. In light of r > 0:5; the rating decisions (30) and (31) imply that fG
fB ; and we consider the space
of parameters that imply fG = 1: We …rst rule out the case of fB = 0: Suppose that fB = 0: The beliefs
expressions (37) - (39) then imply that PrGjGR =PrGjBR = 1 and PrGjN R = 0: It follows from the asset price
expressions (34)-(36) that PGR = PBR = WG and PN R = WB : Substituting for these prices into (31) ; we see
that it is optimal to rate poor quality assets as long as
W > c; which holds by Assumption 2. This implies a
contradiction.
The remaining cases are: (a) fG = fB = 1 and (b) fG = 1 and fB 2 (0; 1): Both cases imply PrGjN R = 0
and therefore PN R = WB .
Consider case (a). For both types of assets to be rated, it must be the case that PGR +(1 r)PBR > WB +c
and (1
r) PGR + rPBR > WB + c: The latter condition is su¢ cient to ensure that both hold. Substituting for
prices and beliefs in that condition and using fB = 1 in the resulting expression, we obtain
c~ <
r(1 r)
)(1 r)][ (1 r) + (1
[ r + (1
)r]
:
Consider case (b). In this case, banks with poor quality assets are indi¤erent between rating and not
rating, i.e.
(1
r) PGR + rPBR
c = WB :
Substituting for prices and beliefs into the above expression, we obtain
(1
r + (1
r) r
)fB (1
r)
+
(1
(1 r) r
= c~:
r) + (1
)fB r
(40)
The mixed strategy fBmix speci…ed in the statement of this lemma is the positive root of the above expression.
r)
Setting fBmix > 0 simpli…es to c~ < 1; while setting fBmix < 1 simpli…es to c~ > [ r+(1 )(1 r(1
r)][ (1 r)+(1
)r] :
Noting that
[ r+(1
r(1 r)
)(1 r)][ (1 r)+(1
)r]
equals 0 at
= 0 and increases in
in the interval
2 (0; 1),
this lemma allows us to characterize the shape of fB ( ) under mandatory ratings disclosure. For low
fB ( ) 2 (0; 1) : As
increases, fB switches to 1 and stays there. For all
2 (0; 1]; we have fB > 0; which we
know implies fG = 1, and therefore no additional restriction is needed to guarantee that fG = 1:
1 4 This
is obtained as a solution to (1
r) PGR + rPBR
c = PN R after substituting for prices and beliefs.
20
,
An important result is formalized in the proposition below. We show that the mandatory ratings disclosure
policy tends to weaken the incentive to screen. The intuition for this result is as follows. The introduction of
this policy no longer allows to pass o¤ missing ratings as undisclosed false ratings. Even poor ratings now serve
as a positive signal. This encourages the rating behavior on the part of poor asset issuers. In fact, it is seen
immediately that fBmix ( ) is strictly greater than its counterpart in the benchmark model (de…ned in Corollary
1) for all : As a result, the amount of market clarity is reduced, just like in the case of the mandatory rating
policy. As the informational value of good ratings declines, so does the expected return to screening, which
compounds the resource misallocation in the economy.
Proposition 4 Macroeconomic Implications of the mandatory ratings disclosure
c~
15
If
2 0; 11 rr r~
the resource misallocation worsens under the mandatory ratings disclosure policy:
c ,
:
md
Proof. See the appendix.
Recall from Corollary 1 that fB ( ) > 0 in the range of
2
0; 11
r c~
r r~
c
: Therefore the result applies
in the empirically relevant cases. Figure 4 illustrates the comparison between the benchmark economy and
the mandatory ratings disclosure economy for the following parameterization: r = 0:8; c = 0:1; WG = 1:1;
WB = 0:3: Panel b shows the rating intensity. Because this is the case where If c > (1
the benchmark rating function fB ( ) is below 1 on the entire range of
discussion, fBmd ( ) > fB ( ) for all
md
k ( ) = (1
, it su¢ ces to show that
0)
r) (2r
1) W ,
: Consistent with our
: This results in the lower premium paid on high quality assets under
mandatory ratings disclosure (panel d) and lower returns to screening
that
2 0;
1 r c~
1 r r~
c
R( ) >
R
md
R ( ) (panel c). Note that to show
( ) in the range of admissible
: This is because
R ( ) and H ( ) increases in k ( ) : Panel a illustrates the actual equilibrium level of good
projects in both economies as the intersection of the 45 degree line with the benchmark H ( ) and Hmd ( ) ;
derived under the assumption that
5.
0
= 0:6 and a linear cumulative distribution function of banks, F (k) = k:
Conclusion
We developed a general equilibrium model that allows us to study the interaction of information production in
secondary markets for loans and screening intensity at the stage of loan origination. The model provides insight
into what determines screening e¤ort at the stage of loan origination and explains why it is less than optimal.
It also shows that screening e¤ort unambiguously weakens as a result of a rise in collateral values, an increase
in the fraction of repaying borrowers, and a decrease in the rating technology precision. The model provides
new insight into several empirical facts that were observed prior to the 2008 …nancial crisis and cautions against
two regulatory policies that have been proposed to be implemented in asset markets, namely, the policies of
mandatory rating and mandatory ratings disclosure. We …nd that, contrary to conventional wisdom, both
policies exacerbate the problem of credit misallocation. Our …ndings suggest that, policy design aimed at
raising rating accuracy (i.e. reducing asset complexity or disciplining the behavior of rating agencies) would be
much more e¤ective at aligning the equilibrium outcome with the e¢ cient one.
1 5 This
condition is more stringent than needed.
21
2
1.9
1.8
1.7
Yield Other / Yield AAA
1.6
1.5
1.4
1.3
1.2
1.1
1
2003
2004
2005
2006
year
2007
2008
2009
Di¤erence in Yields of AAA-rated and Other Tranches
Notes. Source: ABSnet (restricted) deal data. The following restrictions were applied to the dataset.
Region: United States; Deal Type: Asset Backed Security; Asset Class: Small Business Loans. Yield AAA is
computed as the average yield on tranches with at least one triple-A rating. Yield Other is computed as the
average yield on all other tranches.
22
0.3
0.25
0.2
G
B
c / (W -W )
fB = 0
0.15
0 < fB < 1
0.1
fB = 1
0.05
0
0.5
0.55
0.6
0.65
0.7
0.75
r
0.8
0.85
0.9
0.95
1
Equilibrium Rating Strategy
Notes: This …gure helps visualize Lemma 1. For a given resource allocation
, this …gure reports the
optimal rating strategy by holders of type B assets in the space of two exogenous quantities: the rating accuracy
r and the rating cost c normalized by the repayment di¤erential WG
23
WB .
0.83
a. Fraction of Good Proj ects
b. Rating Intensity
1.5
mu*
fB
0.825
1
0.82
0.5
0.815
0
0.81
0.92
0.2
0.91
0.9
r
0.89
0.88
0.92
c. Fraction of Banks Screening
0.91
0.9
r
0.89
0.88
d. Accuracy of a High Rating
1.05
F(k bar )
PrG |G R
0.15
1
0.1
0.95
0.05
0.92
0.8
0.91
0.9
r
0.89
0.9
0.92
0.88
e. Fraction of Assets Rated High
0.9
0.91
0.9
r
0.89
0.88
f. Rating Informativ eness and Price Diff.
PrG |G R- PrG |NR
frac G R
0.78
PG R- P NR
0.8
0.76
0.7
0.74
0.6
0.72
0.7
0.92
0.91
0.9
r
0.89
0.5
0.92
0.88
0.91
0.9
r
0.89
0.88
Equilibrium E¤ects of Decreasing the Rating Precision
Notes: This …gure illustrates the possible equilibrium e¤ects of decreasing the rating accuracy parameter
r:
24
a. Equilibrium Lev el of Good Proj ects
0.7
b. Rating Intensity
2
H( )
H( ), MD
0.69
fB
f B , MD
1.5
0.68
0.67
1
0.66
0.65
0.5
0.64
0
0.63
0.62
-0.5
0.61
0.6
-1
0
0.2
0.4
0.6
0.8
1
0
c. Return to Screening
0.35
0.2
0.4
0.6
0.8
1
0.8
1
d. Price Differential
0.8
P G R-P NR
0.7
0.3
P G R-P BR , MD
0.6
0.25
0.5
0.2
R G -R B
0.4
R G -R B , MD
0.15
0.3
0.1
0.2
0.05
0.1
0
0
0
0.2
0.4
0.6
0.8
1
0
0.2
0.4
0.6
Benchmark Economy vs. Economy under Mandatory Ratings Disclosure
Policy
Notes: This …gure visualizes Proposition 4. For each of the economies, this …gure reports the rating intensity function, premium paid on high quality assets and returns to screening. Panel a shows the determination
of the equilibrium level of good projects in the economy as an intersection of the 45 degree line and function
H ( ):
25
A.
Proof of Lemma 1
Rating strategy fB .
Case 1. fB = 1: From (1) ; this strategy is optimal whenever (1
from the zero pro…t conditions (8) and (9) ; we have
(1
r) W
r) (PGR
PN R ) > c: Substituting for prices
Pr > c;
where Pr PrGjGR PrGjN R : Substituting for beliefs from the consistency conditions (8) and (9) and
from fB = 1 gives the necessary and su¢ cient condition for the above inequality:
(1
r
r) W
r + (1
(1
r)
(1 r)
r)
(1 r) + r(1
(1
)(2r 1)
(2r 1))( (2r 1) + (1
)(1
(r
>
)
c; or
c
:
W
>
r))
(41)
Case 2. fB = 0: Following the same steps as in Case 1, we …nd this strategy is optimal whenever
(1
r) W
Pr < c:
Substituting for beliefs from the consistency conditions and for fB = 0; we obtain
(1
(1 r)
r) + (1
)
1
(1 r)
1 r
r) W 1
<
(1
c; or
<
c
:
W
(42)
Case 3. fB 2 (0; 1): The mixed strategy is optimal whenever
(1
r) W
Pr = c:
Substituting into the above equality for beliefs from (8) and (9) gives
(1
r)
r + (1
r
)fB (1
(1
r)
(1
r) + (1
r)
=
)[(1 fB ) + fB r]
c
:
W
It is straightforward to show that the positive solution to this equation is given by
q
2
2 r (2
c
~
(1
2
r)
+
(1
r)
(~
c+ )
r) (6 4r) c~r
mix
fB =
:
2~
c (1 r) (1
)
(43)
The optimal strategy fB is given by fBmix derived above, but bounded by 0 from below and 1 from above.
Thus we have fB 2 (0; 1) whenever
(1
r)
(r
(1
)(2r 1)
(2r 1))( (2r 1) + (1
r))
<
c
< (1
W
r)
1
1 r
:
The …rst inequality appearing in the expression above is derived by setting fBmix < 1, and the second
inequality is derived by setting fBmix > 0:
Rating strategy fG .
To ensure that fG = 1; we must impose that
r W
Pr > c:
This inequality holds whenever fB > 0; because in that case r W
26
Pr > (1
r) W
Pr
c:
for
However, on the space of parameters that implies fB = 0; we need an additional restriction. Substituting
Pr in the case of fB = 0 and simplifying, we obtain
r W
1
1 r
Pr = r W
> c:
The result follows.
B.
Proof of Proposition 1
Because H( ) is continuous, there exists at least one …xed point of H( ) whenever H(0) > 0 and H ( ) < .
The …xed point is unique if H 0 ( ) < 1 on the entire range of 2 (0; ). The constants 1 ; 2 ; 3 and used in
the proof are de…ned in the statement of Corollary 1.
Existence
Recall the de…nition of H : [0; ] ! [
0 ; 1]
given in (22) :
H( ) = F (k ( )) + 1
F (k ( )
0:
We substitute for the marginal screener
k ( ) = (1
0) [
W
Pr ( ) [r
(1
r) fB ( )]
(1
fB ( )) c]
(44)
using (??) and for market clarity
Pr ( ) =
r + (1
r
)fB ( ) (1
r)
(1
(1 r)
)[(1 fB ( )) + fB ( ) r]
r) + (1
using (8) and (9) :
We use the resulting expression to …nd H (0). We know from Lemma 1 that fB (0) = 0 and fG (0) = 1:
Hence, Pr (0) = 1 and k (0) = (1
c] > 0. It follows that
0) [ W r
H (0) = F ((1
0) [
Wr
c]) + [1
F ((1
0) [
Wr
c])]
0
> F (0) + [1
F (0)]
Our next objective is to …nd H ( ) : By Lemma 1, fB ( ) = 0 and fG ( ) = 1. Hence,
k( )
=
(1
0) [
=
(1
0)
=
(1
0)
=
(1
0)
"
W
Pr ( ) [r (1
1
Wr
c
1 r
Wr
Wr
1
1
c~
r
r
r(1
r
1
c~
c~)
c~
c~
c = (1
r) fB ( )]
(1
0
> 0:
Pr ( ) =
1
1 r
and
fB ( )) c] =
#
c
0 ) [c
c] = 0:
Hence,
H ( ) = F (0) + [1
The assumption
implies that equation
F (0)]
0
=
0:
< then implies that H ( ) < : We showed that H (0) > 0 and H ( ) < ; which
= H ( ) has at least one solution.
0
Uniqueness
Now we move on to discuss uniqueness. Di¤erentiating H ( ) ; we obtain
H 0 ( ) = (1
27
0 )Fk
@k
:
@
Range 1. First, consider the highest admissible range of 2 ( 3 ; ) ; where fB = 0 by Lemma 1. Substituting
r) W
for fB into (18) and di¤erentiating, we obtain @@ k = (1 (10 )r(1
and therefore
r )2
H0 ( ) =
Fk (1
0)
r(1
(1
2
r) W
r )2
0:
Range 2. If parameters satisfy c~ (1 r) (2r 1) ; consider 2 (0; 1 ) and 2 ( 2 ; 3 ) : If instead parameters satisfy c~ > (1 r) (2r 1) ; consider the entire range of 2 (0; 3 ) : By Lemma 1, we have fB 2 (0; 1)
for these values of ; and therefore Pr = (1 r)c W : Substituting this into (18) and di¤erentiating gives
us
@k
@
= 0 and therefore
H 0 ( ) = 0:
Range 3. It remains to consider the range 2 ( 1 ; 2 ) ; relevant only if parameters satisfy c~
By Lemma 1, fB = 1: Substituting that into (18) and di¤erentiating, we obtain
@k
= (1
@
(1
r) (2r
1) :
2
0)
W
r (2r 1) (2
1) (r 1)
,
2
(2r 1)) ( (2r 1) + (1 r))2
(r
which then implies
H0 ( )
=
Fk (1
0)
2
=
Fk (1
0)
2
W
1)2 (1 2 )r(1 r)
1))2 ( (2r 1) + (1
(2r 1)2 (1 2 )
(r
(2r
(2r
r (1
r) 1 + (1
W
)
r 2 +(1 r)2
r(1 r)
r))2
2:
0
H ( ) <
0
H (
1)
1
= f (1
1:
Therefore, we can
given in (17) into the expression for H 0 ( ) given in (45) and noting that
we obtain
c~r(1 r)
1)(1 r c~(2r 1)) ,
= Fk (1
2
0)
(2r 1)2
r(1 r)
W
1+
= Fk (1
(45)
2
The resulting derivative is a decreasing function of ; with a zero at = 0:5:
If
0:5, then H 0 ( ) 0. If < 0:5; then H 0 ( ) > 0; and it is maximized out at
bound H 0 ( ) on the range of 2 ( 1 ; 0:5) by setting H 0 ( 1 ) < 1:
Substituting the expression for
1 (1
1 ) simpli…es to (2r
=
q
(1 r)(2r 1) c~
(2r 1)(~
c+1 2~
cr r)
c~r(1 r)
(2r 1)(1 r c~(2r 1))
r 2 +(1 r)2
r(1 r)
2
=
2
s
(2r 1)2
(1 r) (2r 1) c~
1 r
W
=
0)
r (1 r) (2r 1) (1 r c~ (2r 1))
(1 r c~ (2r 1))
s
2
2
(2r
1)
(1 r) c~= (2r 1)
1 r
2
W
=
< 1;
0)
r (1 r)
(1 r c~ (2r 1))
(1 r c~ (2r 1))
2
2
where f = supk2[0;1] F 0 (k), and the last inequality is satis…ed by the premise.
To summarize, we found that H ( ) is weakly decreasing in the entire range of 2 (0; ) if c~ > (1 r) (2r 1) :
It is also weakly decreasing in the range of 2 (0:5; ) if c~ (1 r) (2r 1) : In all cases, H 0 ( ) < 1; which
ensures that equation = H ( ) has exactly one solution.
It follows that the equilibrium exists, and it is unique.
28
C.
Proof of Lemma 2
From the equilibrium condition
( W ) = H(
( W ); W ); we obtain
H W
@
=
:
@ W
1 H
By Proposition 1, we know that 1 H > 0: Therefore, the sign of
Recalling the de…nition of H from (22),
H( ; W ) = F (k( ; W )) + (1
@
@ W
is determined by the sign of H
F (k( ; W )))
we see that H( ; ) is increasing in W if and only if k( ; ) is increasing in
Recalling the expression for the marginal screener from (18) ;
k( ; W )
=
(1
0 )(RG (
=
(1
0 )f[r
; W)
(1
W.
0;
W.
RB ( ; W ))
r)fB ( ; W )] W
Pr ( ; W )
(1
fB ( ; W )c)g;
note that W enters it through two channels.
First, a higher W implies that the good project repays relatively more, thereby directly raising RG RB
and increasing the incentive to screen. Second, W a¤ects k indirectly by inducing changes in the rating
intensity fB : There are two cases to consider.
Case 1. Suppose that fB ( ) 2 (0; 1). Then W Pr = (1 c r) , and therefore
k( ; W )) = (1
0 )c
2r
1
1
;
r
which is independent of W . It follows that H W = 0; and therefore @@ W = 0:
Case 2. Suppose that fB ( ) 2 f0; 1g. Then fB is constant in the neighborhood of
positive direct e¤ect of W remains. It follows that @@ kW > 0 and @@ W > 0.
D.
, and only the
Proof of Lemma 3
From the equilibrium condition
(r) = H(
(r); r); we obtain
@
Hr
=
:
@r
1 H
By Proposition 1, we know that 1
Recalling the de…nition of H,
H > 0: Therefore, the sign of
H( ; r) = F (k( ; r)) + (1
@
@r
is determined by the sign of Hr .
F (k( ; r)))
0;
we see that H( ; ) is increasing in r if and only if k( ; ) is increasing in r. Recalling the expression for the
marginal screener,
k( ; r)
=
(1
0 )(RG (
=
(1
0 )f(r
; r)
(1
RB ( ; r))
r)fB ( ; r)) W
(46)
Pr ( ; r)
(1
fB )cg;
we see that r enters through two channels. An increase in the rating precision directly increases the payo¤ to
screening by increasing the probability that holders of high quality assets will receive a good rating and sell
their assets at a premium and by decreasing the probability that holders of poor quality assets will receive a
good rating in error and sell at a premium. There is also an indirect e¤ect working through fB which in‡uences
the actual premium paid on a highly rated asset. There are two cases to consider.
29
Case 1. Suppose that fB 2 (0; 1): Then W Pr = (1 c r) : This means that market clarity and hence
the premium paid on high quality assets also increase in r: Both e¤ects work in the same direction. Formally,
@k
1
we substitute for Pr into (46) to obtain k = 2r
1 r c 0 , which is clearly increasing in r: Hence, @r > 0 and
@
@r
> 0:
Case 2. Suppose that fB 2 f0; 1g. Then fB is constant in the neighborhood of
@(RG RB )
= (1 + fB ) W
@r
which implies that
E.
@k
@r
> 0 and
@
@r
Pr +(r
(1
r)fB ) W
@
, and therefore
Pr
> 0;
@r
> 0.
Proof of Lemma 5
From the equilibrium condition
(c) = H(
(c); c); we obtain
Hc
@
=
:
@c
1 H
By Proposition 1, we know that 1
Recalling the de…nition of H,
H > 0: Therefore, the sign of
H( ; c) = F (k( ; c)) + (1
@
@c
is determined by the sign of Hc .
F (k( ; c)))
0;
we see that H( ; ) is increasing in c if and only if k( ; ) is increasing in c.
Recalling the expression for the marginal screener,
k( ; c)
=
(1
0 )(RG (
=
(1
0 )f(r
; c)
RB ( ; c))
(1
r)fB ( ; c)) W
Pr ( ; c)
(1
fB )cg;
we see that c enters through several channels, directly by raising the cost of high quality baskets and indirectly
through fB ( ; c) and Pr ( ; c) : There are three cases to consider.
Case 1. Suppose that fB = 0. Then fB remains constant at 0 in the neighborhood of ; and we have
k = (1
0 )[(2r
1) W
Pr c];
which is strictly decreasing in c, so the result follows.
Case 2. Suppose fB 2 (0; 1): Then W Pr = (1 c r) : Substituting into the above expression, we obtain
2r 1
k = 1 r c 0 , which is increasing in c. The result follows.
Case 3. Suppose that fB = 1. Then fB remains constant at 1 in the neighborhood of , and we have
k = (1
Because
F.
0 )(2r
1) W
Pr :
Pr depends on c only through fB , which is …xed at 1, we have that k is independent of c:
Proof of Lemma 4
From the equilibrium condition
(
0)
= H(
(
@
@
By Proposition 1, we know that 1
0 );
0 );
=
0
we obtain
H
1
0
H
:
H > 0: Therefore, the sign of
30
@
@
0
is determined by the sign of H 0 .
Recalling the de…nition of H,
H( ;
0)
= F (k( ;
0 ))
+ (1
F (k( ;
0 ))) 0 ;
we obtain
@H
@ 0
=
1 + (1
=
(1
@k
@ 0
0 )Fk (k)
F (k)) + (1
F (k)
0 )Fk (k)
Employing the expression for the marginal screener, k( ;
pendent of 0 ; 16 we obtain @@ k =
R:
0)
@k
:
@ 0
= (1
0)
(47)
R and noting that
R is inde-
0
Using it in the derivative in (47) ; we obtain
@H
= (1
@ 0
As a consequence, the following holds.
We have
is weakly increasing in
0
(1
0 )Fk (k)
R:
if
1
0
and
is weakly decreasing in
G.
Proof of Proposition 4
0
F (k))
1
1
;
R
F (k)
Fk (k)
otherwise.
To show that md
, it su¢ ces to show that H md ( )
(1
0 ) R ( ) ; it su¢ ces to show that
Rmd ( ) :
R( )
We will drop the dependence of quantities and prices on
Recall that in the benchmark economy,
R
=
PGR
=
PN R
=
(r
H ( ) : Because H ( ) increases in k ( ) =
for notational clarity.
(1
r) fB ) (PGR PBR ) c (1 fB ) ;
r
W + WB ;
r + (1
)fB (1 r)
(1 r)
W + WB :
(1 r) + (1
)[(1 fB ) + fB r]
1 6 To
see this, consider the following cases. Suppose fB 2 (0; 1), then (27) holds. Using it in the expression for
the e¤ect of fB cancels out, and hence R is independent of 0 :
R
=
=
[r (1 r)fB ] W
(2r 1)c
:
1 r
Pr
(1
R reveals that
fB )c
Alternatively, fB is either 0 or 1. In this case, the e¤ect of 0 through fB is not present because fB is constant. Noting that
Pr is independent of 0 ; we see that R is independent of 0 :
31
In the mandatory disclosure economy,
Rmd
=
md
PGR
=
md
PBR
=
PNmd
R
=
md
md
r) fBmd PGR
+ 1 r fBmd r PBR
r
W + WB ;
r + (1
)fB (1 r)
(1 r)
W + WB ;
(1 r) + (1
)fB r
r
(1
fBmd PNmd
R
1
c 1
fBmd ;
WB :
Case 1. fB ( ) ; fBmd ( ) are both mixed.
In the benchmark economy,
4R = c
obtained by substituting for (PGR PN R ) =
In the mandatory disclosure economy,
4Rmd
md
= rPGR
+ (1
c
1 r
md
r) PBR
=
(2r
md
1) PGR
=
(2r
1) W
2r
1
into
1
;
r
R.
fBmd (1
c
md
md
r) PGR
+ rPBR
c
fBmd PNmd
R
1
md
PBR
c~
(1
r)
(1
r) + (1
;
)fBmd r
md
md
where we used (1 r) PGR
+ rPBR
c = PNmd
Rmd .
R in
md
It follows that 4R < 4R because (1 r)+(1 )f md r > 0:
B
Case 2. fB ( ) = fBmd ( ) = 1.
In the benchmark economy,
4R
=
(r
=
(2r
(1
r
)fB (1
r) fB )
1)
r + (1
r
r + (1
)(1
r)
(1
r)
(1
(1 r)
(1 r) + (1
r) + (1
r)
)(1 fB (1
r))
W
c (1
fB )
W;
)r
obtained by substituting for prices into 4R and using fB = 1 in the resulting expression.
In the mandatory disclosure economy,
4Rmd
=
(r
(1
r) fB ) PGR + (1
=
(2r
1) (PGR
=
(2r
1)
PBR )
r
r + (1
)(1
r
fB r) PBR
r)
(1
(1
(1 r)
r) + (1
f B ) PN R
)r
c (1
fB )
W;
obtained by substituting for the relevant prices into 4Rmr and using fB = 1 in the resulting expression.
We obtain that
4R = 4Rmd :
Case 3. fB ( ) is mixed and fBmd ( ) = 1:
1
We already showed that 4R = c 2r
1 r in the benchmark economy (case 1).
In the mandatory disclosure economy,
4Rmd
=
(2r
1) W
r
r + (1
2
=
(r +
W (2r 1) (1
2r ) (1 (r +
32
)(1
r)
)
;
2r ))
(1
(1 r)
r) + (1
)r
which is a symmetric inverted parabola centered at 0:5.
If c > (1 r) (2r 1) W then the relevant range is
2
4Rmd is maximized at 0:5 where it is valued at (2r 1)
4R =
c (2r 1)
> (2r
(1 r)
1)
2
2 (0; 3 ) de…ned in Corollary 1. In this range of ,
W: It follows that
W = max 4Rmd
2(0;
3)
4Rmd :
If instead c (1 r) (2r 1) W; then this case is in the range of 2 (0; 1 )[( 2 ; 3 ) de…ned in Corollary
1
1. In this range, 4Rmd is maximized at 2 and 3 where it is valued at c 2r
1 r : It follows that
4Rmd
2(0;
max
1 )[(
2;
3)
4Rmd = c
33
2r
1
1
= 4R:
r
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